Electrode Boundary Condition
The electrode boundary condition is one of the most critical components in electric discharge modeling. It is used to specify boundary conditions for electric potential and charge carriers.
Electric Potential Boundary Conditions
The electric potential boundary condition is implemented as a terminal, where you can either prescribe a voltage or connect to an electrical circuit.
Discharge Current
There are two approaches for computing the discharge current (Ref. 5). The first one views the discharge domain as a two-terminal black box. The total power deposited in the domain is given by
where I is the total current and V is the voltage applied between the two terminals.
The total power can also be computed by integrating the power density over the domain:
where J is the total current density, including both conduction and displacement components.
By the principle of energy conservation, P1 = P2. Therefore, the discharge current can be computed as
Another method, more general and used in the electrode feature, involves computing the current flowing through a specific boundary. The current is obtained by boundary integration:
where n is the outward-pointing normal vector. Positive and negative currents correspond to the current flowing into and out of the domain, respectively.
For charge transport models, the conduction current density is computed by summing the current flux contributions from each charge carrier:
General Charge Carrier Boundary Conditions
No Diffusive Flux
In most cases, this is the default boundary condition applied on exterior boundaries and prescribes a vanishing flux due to diffusion across the boundary:
This condition implies that convection flux is included, allowing charge carriers to move across the boundary freely. While this boundary condition does not model the detailed interaction of charge carriers with metal surfaces, it is typically a good starting point for most electric discharges, particularly under high pressure-gap products p·d where the discharge within the domain is more significant than that at the boundary.
Number Density
This condition prescribes the number density of charge carriers at the boundary. The number density cannot be set to zero, as the dependent variables are typically the logarithmic values of the number density. Generally, the number density of positive charge carriers is set to a small value at the anode, while for negative charge carriers, it is set to a small value at the cathode.
Open Boundary
This condition allows charge carriers to drift out of the domain but prevents inward flux. It is particularly useful in models where the cathode and anode are not clearly defined, such as when discharges are driven by AC voltages.
No Flux
This boundary condition prescribes a zero total flux (insulation) across the boundary.
Flux
This condition specifies the total charge carrier flux across a boundary, which could occur due to chemical reactions at the boundary. However, when applicable, it is recommended to use built-in surface emission boundary conditions.
Surface Emission
Surface emission primarily applies to electrons in all media and holes in solids. Three electron emission mechanisms are available.
Secondary Electron Emission
The boundary condition, available for gas domain only, adds secondary electrons generated by the collision of positive ions at the cathode (Ref. 6, p. 71):
where γ is the secondary emission coefficient. This is the mechanism responsible for phenomena such as Trichel pulses. The interface also provides options to define the boundary condition for outward-drifting carriers.
Field Electron Emission
The field electron emission is only available for gas domain and it is computed as:
where the emission current is calculated using the Fowler–Nordheim formula(Ref. 6, p. 69):
where εF is the Fermi energy, φ is the work function non-perturbed by the field, and ξ is a correction factor accounting for the Schottky effect(Ref. 6, p. 70).
Thermionic Emission
The thermionic emission is computed as:
where the emission current is calculated using the Dushman–Richardson formula (Ref. 6, p. 68):
where A0 is the Richardson constant, D is a factor that considers the quantum-mechanical effect. In general, the Schottky effect needs to be considered. This is done by replacing φ (work function or injection barrier energy) with its reduced value.